Electronic Journal of Qualitative Theory of Differential Equations

In this paper, we are concerned with the following magnetic Schrödinger– Poisson system { −(∇+ iA(x))2u + (λV(x) + 1)u + φu = α f (|u|)u + |u|4u, in R3, −∆φ = u2, in R3, where λ > 0 is a parameter, f is a subcritical nonlinearity, the potential V : R3 → R is a continuous function verifying some conditions, the magnetic potential A ∈ Lloc(R 3, R3). Assuming that the zero set of V(x) has several isolated connected components Ω1, . . . , Ωk such that the interior of Ωj is non-empty and ∂Ωj is smooth, where j ∈ {1, . . . , k}, then for λ > 0 large enough, we use the variational methods to show that the above system has at least 2k − 1 multi-bump solutions.


Introduction
In the past few decades, there is a vast literature concerning the nonlinear Schrödinger- where V : R 3 → R is a nonnegative continuous function with inf x∈R 3 V(x) > 0, 1 < p < 5 and ψ : R 3 → C and φ : R 3 → R are two unknown functions. In fact, the first equation in the above system describes quantum (non-relativistic) particles interacting with the electromagnetic field generated by the motion. And φ(x) satisfies the second equation (Poisson equation) in the system, because the potential φ(x) is determined by the charge of wave function itself. Therefore, system (1.1) can be regarded as the coupling of the Schrödinger equation and Poisson equation. If one looks for stationary solutions ψ(x, t) := e −it u(x) of system (1.1), the system can be reduced by −∆u + V(x)u + φ(x)u = |u| p−1 u, in R 3 , In [4], Azzollini and Pomponio considered system (1.2). More precisely, if V is a positive constant, they proved the existence of a ground state solution (u, φ) for 2 < p < 5. If V is a nonconstant potential that is measurable and (possibly) not bounded from below, they obtained a similar existence result for 3 < p < 5. Existence and nonexistence results were also proved when the nonlinearity exhibits a critical growth.
In a celebrated paper [13], by using the variational methods, Ding and Tanaka established multiplicity of multi-bump solutions for a semilinear elliptic equation with deepening potential well. Recently, in [2], Alves and Yang considered system (1.2) which having a general nonlinear term f and assumed the potential V(x) has the form V(x) = λa(x) + 1, where λ is a positive parameter and a : R 3 → R 3 is a nonnegative continuous function. In the interesting paper, the authors proved the existence of positive multi-bump solutions for the system −∆u + (λa(x) For more results on the Schrödinger-Poisson system, we refer the reader to [3, 5, 7, 10, 11, 18, 19, 23-26, 28, 31-34, 36, 38, 40, 41] and the references therein. In recent years, the magnetic nonlinear Schrödinger equation has also received considerable attention where i is the imaginary unit,h is the Planck constant, and A : R N → R N is the magnetic potential. When one looks for standing wave solutions ψ(x, t) := e −iEt/h u(x), with E ∈ R, of the above equation, the problem can be reduced by From a physical point of view, the existence of such solutions and the study of their shape in the semiclassical limit, namely, ash → 0 + is of the greatest importance, since the transition from Quantum Mechanics to Classical Mechanics can be formally performed by sending the Planck constanth to zero. As far as we know, the first result involving the magnetic field was obtained by Esteban and Lions [15]. In [15], forh > 0 fixed and special classes of magnetic fields, the authors found the existence of standing waves to problem (1.3) by solving an appropriate minimization problem for the corresponding energy functional in the cases of N = 2 and 3. Afterwards, in [27], Kurata assumed a technical condition relating V(x) and A(x). Under these assumptions, he proved that the associated functional satisfies the Palais-Smale condition at any level and further obtained a least energy solution of the problem for any > 0. Also, Alves et al.
[1] studied the multiple solutions by combining a local assumption on V, the penalization techniques of del Pino and Felmer [12] and the Ljusternic-Schnirelmann theory.
Recently, Tang [35] considered multi-bump solutions of the following nonlinear magnetic Schrödinger equation with critical frequency where λ > 0, E ∈ R is a constant, inf x∈R N V(x) = E and f satisfies subcritical growth. Later, by using the variational methods, Ji and Rȃdulescu [22] established the existence and multiplicity of multi-bump solutions for the following nonlinear magnetic Schrödinger equation where λ > 0, f (t) is a continuous function with exponential critical growth, the magnetic potential A : R 2 → R 2 is in L 2 loc R 2 , R 2 and the potentials V, Z : R 2 → R are continuous functions verifying some conditions. Recently, Ma and Ji [30] studied the existence and multiplicity of multi-bump solutions for the magnetic Schrödinger-Poisson system with subcritical growth. It is natural to consider multiplicity of multi-bump solutions for the magnetic Schrödinger-Poisson system with critical growth. To the best of our knowledge, this problem has not ever been studied. For more results related to the nonlinear partial differential equations with magnetic field, we refer to [6,8,9,14,17,20,21,39,42] and references therein.
Inspired by the previous works of [22,30,35], the aim of this paper is to study existence of multi-bump solutions for the magnetic Schrödinger-Poisson system with critical growth where λ > 0 is a parameter, the magnetic potential A is in L 2 loc (R 3 , R 3 ), f has subcritical growth and the potential V : R 3 → R is continuous. Due to the appearance of magnetic field A(x), problem (1.4) can not be changed into a pure real-valued problem, hence we should deal with a complex-valued directly. Also, since the electrostatic potential φ(x) depends on the wave function, φ(x)u is nonlocal which will make some estimates more difficult and complicated. Moreover, since the problem we deal with has critical growth, we need more refined estimates to overcome the lack of compactness. Now we present the general assumptions on the potentials in this paper: nonempty bounded open subset with smooth boundary and Ω = V −1 (0) where int V −1 (0) denotes the set of the interior points of V −1 (0), Ω consists of k components: and Ω i ∩ Ω j = ∅ for all i = j.
The paper is organized as follows. In Section 2, we shall introduce the variational setting and give some necessary preliminaries. In Section 3, we study an modified problem, and prove the Palais-Smale condition for the modified problem and study the behavior of (PS) ∞ sequence. Moreover, we establish L ∞ estimate of the solution of the modified problem. In Section 4, by adapting the deformation flow method, we show that the existence of a special critical point and prove the main theorem.

Preliminaries
In this section, we shall present the variational framework for problem (1.4) and some useful preliminary lemmas.
For u : R 3 → C, let us denote by C is an Hilbert space under the scalar product where Re and the bar denote the real part of a complex number and the complex conjugation, respectively. Moreover, the norm induced by the product ·, · is u A = R 3 |∇ A u| 2 +|u| 2 dx 1 2 . By (A), on H 1 A R 3 , C , we have the important diamagnetic inequality (see [29], Theorem 7.21) which is frequently used in this paper: For λ ≥ 0, a direct computation gives that (E λ , · λ ) is an Hilbert space and E λ ⊂ H 1 A R 3 , C . Also, for an open set K ⊂ R 3 , Let H 0,1 A (K, C) be the Hilbert space obtained as the closure of C ∞ 0 (K, C) under the norm u H 1 A (K,C) . The diamagnetic inequality (2.1) implies that, if u ∈ H 1 A R 3 , C , then |u| ∈ H 1 R 3 , R and u ≤ u A . Therefore, the embedding H 1 A R 3 , C → L r R 3 , C is continuous for 2 ≤ r ≤ 6 and the embedding H 1 A R 3 , C → L r loc R 3 , C is compact for 1 ≤ r < 6. By the continuous embedding H 1 R 3 , R → L r R 3 , R for 2 ≤ r ≤ 6, we have For any u ∈ H 1 A R 3 , C , we obtain that |u| ∈ H 1 R 3 , R , and the linear functional L |u| : is well defined and continuous in view of the Hölder inequality and (2.2). Indeed, we can see that Then, given u ∈ H 1 A R 3 , C , |u| ∈ H 1 R 3 , R , by the Lax-Milgram Theorem, there exists an unique φ = φ |u| ∈ D 1,2 R 3 , R such that −∆φ = u 2 .
Moreover, φ |u| can be expressed as Next, we provide the following properties about φ |u| in the following lemma whose proof is similar to one in [11,32,41], so we omit it.
Now, we define the energy functional I λ associated with problem (1.4) given by it is standard to prove that I λ (u) ∈ C 1 (E λ , R), and for any ϕ ∈ E λ , we have for all u ∈ E λ (K), and λ > 0, where u 2 2,K = K |u| 2 dx. So, from this relation, we have the following result: , and λ > 0.

A modified problem
Since R 3 is unbounded and nonlinear term has the critical growth, we know that the Sobolev embeddings are not compact, as so I λ can not verify the Palais-Smale condition. In order to overcome this difficulty, we adapt the argument of the penalization method introduced by del Pino and Felmer [12] and Ding and Tanaka [13], and consider a modified problem satisfying the Palais-Smale condition. Also, Next, we fix a non-empty subset Γ ⊂ {1, . . . , k} and Using the above notations, we set the functions In view of ( f 1 )-( f 4 ), we have that g is a Carathéodory function satisfying the following properties: is strictly increasing in t ∈ (0, +∞) and for each x ∈ R 3 \ Ω Γ , the function t → g(x,t) t is strictly increasing in (0, a).
Moreover, we have the modified problem We want to get some nontrivial solutions of (3.3) are ones of the original problem (1.4), more precisely, if u λ is a nontrivial solution of (3 Next, we prove that the energy functional Φ λ (u) satisfies the (PS) condition.
On the other hand, by (g 4 ), (g 5 ), κ > θ θ−2 , and Lemma 2.3, we derive For each fixed j ∈ Γ, let us denote by c j the minimax level of the functional I j : H 0,1 A (Ω j , C) → R given by and It is well-known that the critical points of are the weak solutions of the problem Moreover, we have the following important result.
The above inequality implies that .
The above inequality implies that .
From Lemma 3.1, we know that the sequence (u n ) is bounded in E λ . Thus, there exists u ∈ E λ such that u n u in E λ , up to a subsequence if necessary. Then it is standard to check that for any Form (3.7), the density of C ∞ 0 R 3 , C in E λ , and Φ λ (u n ) → 0, we can obtain that the weak limit u is a critical point of Φ λ and so On the other hand, we know that < Φ λ (u n ) , u n >= o n (1) which implies that g(x, |u n | 2 )|u n | 2 dx + o n (1). (3.9) Step 1: We show that for any given ζ > 0, there exists R > 0 large enough such that Ω Γ ⊂ B R/2 (0) and where C > 0 is a constant independent of R. By a direct computation, we have Notice that |Re (u n ∇ A u n )| = |Re ((∇u n + iAu n ) u n )| = |Re (u n ∇u n )| = |u n | |∇|u n ||.
Using the Hölder inequality and the above equality, we derive So, we obtain which implies that for any ζ > 0, choosing a R > 0 larger if necessary, we have lim sup Step 2: We show that By (3.10) and the Sobolev embedding, for any ζ > 0, there exists R > 0 such that for n large enough and q ∈ [2, 6) which implies u n → u in L q (R 3 , C), ∀q ∈ [2, 6).
Hence, by (3.16) and (3.18), lim n R 3 g(x, |u n | 2 )|u n | 2 dx = R 3 g(x, |u| 2 )|u| 2 dx. Using (3.10) and the diamagnetic inequality (2.1), the sequence (|u n |) is tight in, we may assume that |∇|u n || 2 µ and |u n | 6 ν (3.19) in the sense of measures. By the concentration-compactness principle in [37], we can find an at most countable index I, sequences ( for any i ∈ I, where δ x i is the Dirac mass at the point x i . Let us show that (x i ) i∈I ∩ Ω Γ = ∅. Assume, by contradiction, that x i ∈ Ω Γ for some i ∈ I. For any ρ > 0, we define Using the diamagnetic inequality (2.1) again, it follows that Due to the fact that f has the subcritical growth and ψ ρ has the compact support, we have that Because of the boundedness of (u n ) in E λ , using the Hölder inequality, the strong convergence of (|u n |) in L 2 loc (R 3 , R), |u| ∈ L 6 (R 3 , R), |∇ψ ρ | ≤ Cρ −1 and |B 2ρ (x i )| ∼ ρ 3 , we have that (3.24) Now, from ( f 3 ), (g 4 ) and (g 5 ), we have From the above arguments, (3.20) and (3.24), we have which gives a contradiction. This means that (3.17) holds. From (3.8), (3), (3.12) and (3.13), we may obtain that u n 2 λ → u 2 λ which means that u n → u in E λ .
Next we study the behavior of a (PS) ∞ sequence, that is, a sequence (u n ) ⊂ H 1 A R 3 , C satisfying u n ∈ E λ n and λ n → ∞, ). Then, up to a subsequence, there exists u ∈ H 1 A R 3 , C such that u n u in H 1 A R 3 , C . Moreover, (iv) u n 2 λ n ,Ω j → Ω j (|∇ A u| 2 + |u| 2 )dx, for j ∈ Γ; (v) u n 2 λ n ,R 3 \Ω Γ → 0; Proof. As in Lemma 3.1, we know that (u n ) is bounded in H 1 A (R 3 , C). Thus we may assume that for some u ∈ H 1 A (R 3 , C), up to a subsequence, if necessary By the Fatou's lemma, we derive C m |u| 2 dx = 0.
From ( f 1 ), ( f 2 ), for any ζ > 0, there exists C ζ > 0 such that So, we derive Therefore, Since for each v ∈ C ∞ 0 Ω j , C , Φ λ n (u n ) v → 0 as n → ∞, from the above information and the argument explored in Proposition 3.4, we have Re which implies that u| Ω j is a solution of problem (3.25) for each j ∈ Γ.
Proposition 3.6. Let (u λ ) be a family of nontrivial solutions of (3.3). Then, there exists λ * > 0 such that u λ 2 L ∞ (R 3 \Ω Γ ) ≤ a, ∀λ ≥ λ * . In particular, u λ is a solution of the original problem (1.4) for any λ ≥ λ * . Proof. We give notation B r (x) = y ∈ R 3 : |x − y| < r . Since u λ ∈ E λ is a critical point of Φ λ (u), that is, u λ satisfies the following equation By the Kato's inequality We use the subsolution estimate (see [16], Theorem 8.17) and obtain that there exists a constant C(r) such that for 1 < q < 2 sup y∈B r (x) |u λ | q dy 1/q . By Proposition 3.5, for any sequence λ n → ∞, we can extract a subsequence λ n i such that In particular, Since λ n → ∞ is arbitrary, we have Thus, choosing r ∈ (0, dist(Ω Γ , R N \ Ω Γ )), we have uniformly in x ∈ R N \ Ω Γ that This finishes the proof.

Existence of multi-bump solutions
In this section, we start to prove the existence of multi-bump solutions. First of all, for each fixed j ∈ Γ, let us denote by c j the minimax level of the functional I j : H 0,1 A Ω j , C → R given by For each j ∈ Γ, we denote by Φ λ,j : H 1 A Ω j , C → R the functional given by and the above functional is associated to the following problem In what follows, we denote by c λ,j the minimax level of the above functional given by Repeating the same method used in the previous section, we are able to prove that there exist ω j ∈ H 0,1 A Ω j , C and ω λ,j ∈ H 1 A Ω j , C such that I j (ω j ) = c j and I j (ω j ) = 0, and Φ λ,j (ω λ,j ) = c λ,j and Φ λ,j (ω λ,j ) = 0.
Proof. (i) From ( f 3 ), we have c j > 0 and c λ,j > 0 for any j ∈ Γ and λ ≥ 1. For any u ∈ H 0,1 A Ω j , C , we may extend u to u ∈ H 1 A Ω j , C by Using the fact that H 0,1 1] I j (γ(t)) = c j .
(ii) By the monotonicity of the term f (t) with respect to t for t > 0, we are able to prove this.
By the definition of c j , we have Together with (i), we get the asserted result.
In what follows, we fix R > 1 verifying and By the definition of c j , we are able to obtain max s j ∈[1/R 2 ,1] I j s j Rω j = c j , ∀j ∈ Γ.
This shows what was stated.
(iv) By (ii), we can choose λ large enough such that b λ,Γ , c Γ ∈ (0, 1 3 S 3 2 ). From Proposition 3.4 and (3.6), we know that any (PS) b λ,Γ sequence of Φ λ has a convergence subsequence in E λ . Moreover, from the deformation lemma, we can conclude that b λ,Γ is a critical level of Φ λ for λ large.
To prove Theorem 1.1, we need to find a nontrivial solution u λ for the large λ which approaches a least energy solution in each Ω j (j ∈ Γ) and to 0 elsewhere as λ → ∞. Therefore, we shall show two propositions which imply together with the estimates made in the previous section that Theorem 1.1 holds.
Henceforth, let We obtain the following uniform estimate of Φ λ (u) λ on the annulus Proof. Arguing by contradiction, we assume that there exist λ n → ∞ and u n ∈ A λ n 2µ Since u n ∈ A λ n 2µ , we can obtain that u n λ n is a bounded in E λ n R 3 , C and H 1 A R 3 , C , and {Φ λ n (u n )} is also bounded. Thus, passing a subsequence if necessary, we may assume that From Proposition 3.5, there exists u ∈ H 0,1 A (Ω Γ , C) such that u is a solution of (3.25), Since c Γ = ∑ l j=1 c j and c j is the least energy level for I j , we have two possibilities: (i) I j (u| Ω j ) = c j ∀j ∈ Γ; (ii) I j 0 (u| Ω j 0 ) = 0, that is u| Ω j 0 ≡ 0 for some j 0 ∈ Γ.
Thus, Φ λ,j (u) − c j ≤ µ, ∀j ∈ Γ, that is, u n ∈ A λ n µ for large n, which is a contradiction to the assumption u n ∈ A λ n 2µ \ A λ n µ . If (ii) occurs, we have Φ λ n ,j 0 (u n ) − c j 0 → c j 0 ≥ 3µ, which is a contradiction with the fact that u n ∈ A λ n 2µ \ A λ n µ . Thus neither (i) nor (ii) can hold, and the proof is completed.